Résumé
In this paper, we consider a mathematical model of two herbivore groups and two available resources, described by a control system. The controls model how the herbivores feed on the two types of resources. The choice of these controls is based on the standard assumption of optimal feeding theory that requires each predator to maximize the net rate of energy intake during feeding. Qualitative analysis of the model highlights four resource thresholds, called resource switching densities. At these thresholds, herbivores can have pure dynamics, meaning that they consume only one type of resource, or adaptive dynamics, meaning that they consume both types of resources. Various equilibria, including the coexistence equilibrium and boundaries equilibria, are computed and their stability analysis are rigorously studied. A detailed description of the switching zones is made using Filippov’s solutions. Notably, we find that within the framework of adaptative dynamics, the system components may experience a long term sustainable coexistence where in the pure dynamics case, at least one component may go to extinction. Also, thanks to the adaptative dynamics, periodic behaviors in the system are observed. Finally, we provide some numerical simulations in order to illustrate our qualitative results